Of miracles (1748)
P(deception) >> P(resurrection)
Mémoire sur la probabilité des causes par les événements (1774)
Direct and inverse (conditional) probabilities
\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
\(P(B|A) = \frac{P(B \cap A)}{P(A)}\)
Procedure:
\[\begin{align} P(B|A)P(A) &= P(B \cap A)\\ P(B|A)P(A) &= P(A \cap B)\\ P(A|B) &= \frac{P(B|A)P(A)}{P(B)}\\ P(A|B) &= P(A)\frac{P(B|A)}{P(B)} \end{align}\]
\(posterior = prior \times
normalized\;likelihood\)
\(\textrm{Update Factor} = P(B|A) / P(B)\)
\[\begin{align} P(B|A)/P(B) &> 1\\ P(A|B) &> P(A) \end{align}\]
\(P(H|E) = P(H)\frac{P(E|H)}{P(E)}\)
where:
\[\begin{align} P(H/E) &= \textrm{Probability of the hypothesis, given the evidence (Posterior)}\\ P(H) &= \textrm{Probability of the hypothesis, before getting the evidence (Prior)}\\ P(E|H) &= \textrm{Probability of the evidence, given the hypothesis (Likelihood)}\\ P(E) &= \textrm{Probability of the evidence (Marginal)} \end{align}\]